The list Unfriendly Partition Conjecture for countable graphs

Let $G$ be a countable graph, and assign to each vertex $v$ a list $L(v)$ of available colors. The graph is majority $k$-choosable if, for every such assignment with $|L(v)|=k$, it has a majority coloring using a color from $L(v)$ at each vertex, where at each vertex the number of incident monochromatic edges is at most the number of incident edges whose endpoints have different colors. List Unfriendly Partition Conjecture. Every countable graph is majority $2$-choosable. This strengthens the Unfriendly Partition Conjecture. The paper proves the weaker bound that every countable graph is majority $4$-choosable, so the conjectured bound remains open.